Solve 6x^2+37x-244

$$6 { x }^{ 2 } +37x-244$$

Factor

$$\left(x-4\right)\left(6x+61\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-244. To find a and b, set up a system to be solved.

$$a+b=37 ab=6\left(-244\right)=-1464$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -1464.

$$-1,1464 -2,732 -3,488 -4,366 -6,244 -8,183 -12,122 -24,61$$

Calculate the sum for each pair.

$$-1+1464=1463 -2+732=730 -3+488=485 -4+366=362 -6+244=238 -8+183=175 -12+122=110 -24+61=37$$

The solution is the pair that gives sum 37.

$$a=-24 b=61$$

Rewrite 6x^{2}+37x-244 as \left(6x^{2}-24x\right)+\left(61x-244\right).

$$\left(6x^{2}-24x\right)+\left(61x-244\right)$$

Factor out 6x in the first and 61 in the second group.

$$6x\left(x-4\right)+61\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(6x+61\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$6x^{2}+37x-244=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-37±\sqrt{37^{2}-4\times 6\left(-244\right)}}{2\times 6}$$

Square 37.

$$x=\frac{-37±\sqrt{1369-4\times 6\left(-244\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-37±\sqrt{1369-24\left(-244\right)}}{2\times 6}$$

Multiply -24 times -244.

$$x=\frac{-37±\sqrt{1369+5856}}{2\times 6}$$

Add 1369 to 5856.

$$x=\frac{-37±\sqrt{7225}}{2\times 6}$$

Take the square root of 7225.

$$x=\frac{-37±85}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-37±85}{12}$$

Now solve the equation x=\frac{-37±85}{12} when ± is plus. Add -37 to 85.

$$x=\frac{48}{12}$$

Divide 48 by 12.

$$x=4$$

Now solve the equation x=\frac{-37±85}{12} when ± is minus. Subtract 85 from -37.

$$x=-\frac{122}{12}$$

Reduce the fraction \frac{-122}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{61}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and -\frac{61}{6} for x_{2}.

$$6x^{2}+37x-244=6\left(x-4\right)\left(x-\left(-\frac{61}{6}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$6x^{2}+37x-244=6\left(x-4\right)\left(x+\frac{61}{6}\right)$$

Add \frac{61}{6} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6x^{2}+37x-244=6\left(x-4\right)\times \frac{6x+61}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6x^{2}+37x-244=\left(x-4\right)\left(6x+61\right)$$

Evaluate

$$\left(x-4\right)\left(6x+61\right)$$